Find the linear nearminimax approximation for fx ex on 1 1

Find the linear near-minimax approximation for f(x) = e^x on [-1, 1]. Plot your approximation against function.

Solution

Ans-

The Chebyshev polynomials of the first and second kinds are well known. In the case of a real variable x on [ - 1, 11, they are defined by T,(x) = cos no, (1.1) u,(x) = sin(n + l)O sin 8 ’ (1.2) Correspondence to: Dr. G.H. Elliott, School of Mathematical Studies, University of Portsmouth, Hampshire Terrace, PO1 2EG, United Kingdom. 0377-0427/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved 292 J.C. Mason, G.H. Elliott / Near-minimax complex approximation where the subscript n denotes the polynomial, degree, and where x = cos 8. However, real Chebyshev polynomials of the third and fourth kinds may also be defined, and the relevant formulae are Y&4 = cos( n + $)e cos(@) ’ K(x) = sin( n + +)O sin(@) ’ (1.3) (1.4) These latter polynomials have appeared in various guises in the literature. They have been called “airfoil polynomials”, since they are appropriate for approximating the single square root singularities that occur at the sharp end of an airfoil, and a discussion and set of references may be found in [3]. They are also sometimes referred to as “half angle shifted” Chebyshev polynomials. However their apt designation as third and fourth kind Chebyshev polynomials is apparently due to Gautschi [4], in consultation with colleagues in the field of orthogonal polynomials. In fact the four polynomials (l.l)-(1.4) form a natural quartet, since they are Jacobi polynomials for the four parameter pairs LY = f i, p = + i, being orthogonal on [ - 1, 11 with respect to (1 -x’)-“~, (1 --~~)i’~, (1 -~)r’~(l +.X)-I/~ and (1 +n)“‘(l -x)-i”. The four kinds of Chebyshev polynomial are all readily generated from their common recurrence relation P,(X) = 2%-,(X) -P,-2(X)> PO(X) = 19 (1.5) but with differing choices of starting values p*(x) =x, 2x, 2x - 1 and 2x + 1, respectively. (1.6) In two recent papers of the present authors, some new results and properties were obtained on the real interval [ - 1, 11. We considered classes C, r[ - 1, I], C_ i[ - 1, 11 and C, i[ - 1, 11 of functions continuous on [ - 1, l] but vanishing at f 1, - 1 or + 1, respectively [9]. We showed that expansions in {7’,(x)}, {(l -x*)‘/*U,(x>}, {(l +x>‘/*V,Jx)) and {(l -x)‘/~W,(X>} yielded near-minimax approximations in C[ - 1, 11, C + i[ - 1, 11, C_,[- 1, 11 and C+l[- 1, 11, respectively, and obtained explicit formulae for the- norms of the associated projections. We also obtained corresponding results for interpolation at Chebyshev polynomial zeros. The first author [7] discussed the application of the four kinds of Chebyshev polynomials to certain problems involving indefinite integration or integral transforms. The main purpose of the present paper is to seek complex near-minimax series expansion results analogous to the real results of [9], by extending the definitions of Chebyshev polynomials appropriately to a complex variable z. However, several key differences arise in the complex case. Firstly, an elliptical domain is involved. Secondly, results depend on the size of the domain. Thirdly, analytic rather than continuous function spaces are the more natural setting. Fourthly, weighted rather than constrained function spaces are adopted. Finally, it is not apparently possible to provide attainable bounds for the norms of series projections. The new results establish that weighted Chebyshev series expansions provide excellent (near-minimax) approximations to weighted analytic functions around square root singularities,